Adiabatic dressing of quantum enhanced Markov chains

Abstract

Quantum-enhanced Markov chain Monte Carlo, a hybrid quantum-classical algorithm in which configurations are proposed by a quantum proposer and accepted or rejected by a classical algorithm, has been introduced as a possible method for robust quantum speedup. Previous work has identified competing factors that limit the algorithm's performance: the quantum dynamics should delocalize the system across a range of classical states to propose configurations beyond the reach of simple classical updates, whereas excessive delocalization produces configurations unlikely to be accepted, slowing the chain's convergence. Here, we show that controlling the degree of delocalization by adiabatically dressing the quench protocol can significantly enhance the Markov gap in paradigmatic spin-glass models.

Figures (9)

• Figure 1: Three-stage ramp protocol for the Hamiltonian parameter $\gamma(t)$. A smooth ramp-up from 0 to 1 over a duration $\alpha$ is defined by Eq. \ref{['eq:ramp']}, followed by a plateau of length $\kappa$, and a ramp-down obtained by time-reversing the ramp-up.
• Figure 2: Bottleneck bound on the spectral gap $\delta$ for sampling the Ising chain at $\beta=5$ and fixed transverse field $h=1.5$. Solid lines show the bound for $N=8-40$, while dots denote exact gaps for smaller systems. The inset shows the system-size dependence of the peak gap, exhibiting polynomial scaling.
• Figure 3: Spectral gap of the adiabatically dressed MCMC protocol at $\beta = 5$ and $h=1.5$ for the SK (green) and 3-spin (purple) models. Panels (a) and (c) show the disordered-averaged gap $\langle \delta\rangle$, as a function of the ramp time $\alpha$ for $N=5-12$, with lighter to darker shades indicating increasing system size. Blue triangles denote the corresponding quench gaps, and black dots mark the peak gaps. Panels (b) and (d) show the dependence of the gap on the plateau duration $\kappa$, evaluated at the gap-maximizing $\alpha$. Dashed horizontal lines indicate the gap averaged over $\kappa \in [10^{2},10^{5}]$. The insets in (a) and (c) show the system-size dependence of the peak gap (black circles), the $\kappa$-averaged gap (squares), and the quench gap (blue triangles). Fits to the exponential scaling of the inset data, reported in Table \ref{['tab:fits_k']}, demonstrate improved scaling relative to the quench strategy for both models.
• Figure A1: The Hamiltonian parameter $\gamma(t)$ is linearly ramped from $0$ to $1$ over a duration $\alpha$ is followed by a plateau of length $\kappa$ and a time-reversed ramp back to $0$. Spectral gaps using this protocol in the large-$\kappa$ limit are shown in Fig. \ref{['fig:linearramp']}.
• Figure A2: Spectral gap of the linear ramp protocol versus ramp time for the SK (a) and 3-spin (b) model at fixed transverse field $h=1.5$, showing similar behaviour to that in Fig. \ref{['fig:combine']}. The inset shows the system-size dependence of the peak gap (black circles), from which the exponential scaling exponents $k_{peak}=0.129(3)$ for the SK model and $k_{peak}=0.146(3)$ for the 3-spin model are extracted.